An important feature of SMC methods is that unbiased estimates of marginal likelihoods are available, making the pseudo-marginal approach by [1]} possible. Given a current parameter \(\mathbf {\theta }\)  with corresponding estimate of the marginal likelihood \(\hat{p}(\mathbf {y}_{1:T};\mathbf {\theta })\) , a proposal \(\tilde{\mathbf {\theta }}\sim q(\tilde{\mathbf {\theta }}|\mathbf {\theta })\)  is generated and an estimate \(\hat{p}(\mathbf {y}_{1:T};\tilde{\mathbf {\theta }})\)  is obtained by a new run of the SMC algorithm. The new proposal will then be accepted with probability
\(\min \lbrace 1,\frac{p(\tilde{\mathbf {\theta }})\hat{p}(\mathbf {y}_{1:T};\tilde{\mathbf {\theta }})q(\mathbf {\theta }|\tilde{\mathbf {\theta }})}{p(\mathbf {\theta }\hat{)} p(\mathbf {y}_{1:T};\mathbf {\theta })q(\tilde{\mathbf {\theta }}|\mathbf {\theta })}\rbrace ,\) 
